Geodesics on an ellipsoid

N

E

A geodesic on an oblate ellipsoid

The study of geodesics on an ellipsoid arose in con-nection with geodesy specically with the solution oftriangulation networks. The gure of the Earth is wellapproximated by an oblate ellipsoid, a slightly attenedsphere. A geodesic is the shortest path between two pointson a curved surface, i.e., the analogue of a straight line ona plane surface. The solution of a triangulation networkon an ellipsoid is therefore a set of exercises in spheroidaltrigonometry (Euler 1755).If the Earth is treated as a sphere, the geodesics are greatcircles (all of which are closed) and the problems re-duce to ones in spherical trigonometry. However, Newton(1687) showed that the eect of the rotation of the Earthresults in its resembling a slightly oblate ellipsoid and,in this case, the equator and the meridians are the onlyclosed geodesics. Furthermore, the shortest path betweentwo points on the equator does not necessarily run alongthe equator. Finally, if the ellipsoid is further perturbedto become a triaxial ellipsoid (with three distinct semi-axes), then only three geodesics are closed and one ofthese is unstable.The problems in geodesy are usually reduced to two maincases: the direct problem, given a starting point and an ini-tial heading, nd the position after traveling a certain dis-tance along the geodesic; and the inverse problem, giventwo points on the ellipsoid nd the connecting geodesicand hence the shortest distance between them. Becausethe attening of the Earth is small, the geodesic distancebetween two points on the Earth is well approximated by

Isaac Newton

the great-circle distance using the mean Earth radiusthe relative error is less than 1%. However, the courseof the geodesic can dier dramatically from that of thegreat circle. As an extreme example, consider two pointson the equator with a longitude dierence of 17959;while the connecting great circle follows the equator, theshortest geodesics pass within 180 km of either pole (theattening makes two symmetric paths passing close to thepoles shorter than the route along the equator).Aside from their use in geodesy and related elds suchas navigation, terrestrial geodesics arise in the study ofthe propagation of signals which are conned (approxi-mately) to the surface of the Earth, for example, soundwaves in the ocean (Munk & Forbes 1989) and the radiosignals from lightning (Casper & Bent 1991). Geodesicsare used to dene some maritime boundaries, which inturn determine the allocation of valuable resources assuch oil and mineral rights. Ellipsoidal geodesics alsoarise in other applications; for example, the propagationof radio waves along the fuselage of an aircraft, whichcan be roughly modeled as a prolate (elongated) ellipsoid(Kim & Burnside 1986).

1

2 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

Leonhard Euler

Geodesics are an important intrinsic characteristic ofcurved surfaces. The sequence of progressively morecomplex surfaces, the sphere, an ellipsoid of revolution,and a triaxial ellipsoid, provide a useful family of sur-faces for investigating the general theory of surfaces. In-deed, Gausss work on the survey of Hanover, which in-volved geodesics on an oblate ellipsoid, was a key moti-vation for his study of surfaces (Gauss 1828). Similarly,the existence of three closed geodesics on a triaxial ellip-soid turns out to be a general property of closed, simplyconnected surfaces; this was conjectured by Poincar(1905) and proved by Lyusternik & Schnirelmann (1929)(Klingenberg 1982, 3.7).

1 Geodesics on an ellipsoid of rev-olution

There are several ways of dening geodesics (Hilbert &Cohn-Vossen 1952, pp. 220221). A simple denitionis as the shortest path between two points on a surface.However, it is frequently more useful to dene them aspaths with zero geodesic curvaturei.e., the analogue ofstraight lines on a curved surface. This denition en-compasses geodesics traveling so far across the ellipsoidssurface (somewhat less than half the circumference) thatother distinct routes require less distance. Locally, thesegeodesics are still identical to the shortest distance be-tween two points.By the end of the 18th century, an ellipsoid of revolution

(the term spheroid is also used) was a well-accepted ap-proximation to the gure of the Earth. The adjustmentof triangulation networks entailed reducing all the mea-surements to a reference ellipsoid and solving the result-ing two-dimensional problem as an exercise in spheroidaltrigonometry (Bomford 1952, Chap. 3).

s121

2

A

B

12

N12

0E F H

Fig. 1. A geodesic AB on an ellipsoid of revolution. N is thenorth pole and EFH lie on the equator.

It is possible to reduce the various geodesic problems intoone of two types. Consider two points: A at latitude 1and longitude 1 and B at latitude 2 and longitude 2(see Fig. 1). The connecting geodesic (from A to B) isAB, of length s12, which has azimuths 1 and 2 at thetwo endpoints.[1] The two geodesic problems usually con-sidered are:

1. the direct geodesic problem or rst geodesic problem,given A, 1, and s12, determine B and 2;

2. the inverse geodesic problem or second geodesicproblem, given A and B, determine s12, 1, and 2.

As can be seen from Fig. 1, these problems involve solv-ing the triangle NAB given one angle, 1 for the directproblem and 12 = 2 1 for the inverse problem, andits two adjacent sides. In the course of the 18th centurythese problems were elevated (especially in literature inthe German language) to the principal geodesic problems(Hansen 1865, p. 69).For a sphere the solutions to these problems are simple ex-ercises in spherical trigonometry, whose solution is givenby formulas for solving a spherical triangle. (See the ar-ticle on great-circle navigation.)For an ellipsoid of revolution, the characteristic constantdening the geodesic was found by Clairaut (1735). Asystematic solution for the paths of geodesics was givenby Legendre (1806) and Oriani (1806) (and subsequentpapers in 1808 and 1810). The full solution for the di-rect problem (complete with computational tables and aworked out example) is given by Bessel (1825).[2]

Much of the early work on these problems was carried outby mathematiciansfor example, Legendre, Bessel, and

1.1 Equations for a geodesic 3

Alexis Clairaut

Barnaba Oriani

Gausswho were also heavily involved in the practicalaspects of surveying. Beginning in about 1830, the dis-ciplines diverged: those with an interest in geodesy con-centrated on the practical aspects such as approximationssuitable for eld work, while mathematicians pursued thesolution of geodesics on a triaxial ellipsoid, the analysisof the stability of closed geodesics, etc.During the 18th century geodesics were typically referredto as shortest lines.[3] The term geodesic line was

coined by Laplace (1799b):

Nous dsignerons cette ligne sous le nomde ligne godsique [We will call this line thegeodesic line].

This terminology was introduced into English either asgeodesic line or as geodetic line, for example (Hutton1811),

A line traced in the manner we have nowbeen describing, or deduced from trigonomet-rical measures, by the means we have indi-cated, is called a geodetic or geodesic line: ithas the property of being the shortest whichcan be drawn between its two extremities onthe surface of the Earth; and it is therefore theproper itinerary measure of the distance be-tween those two points.

In its adoption by other elds geodesic line, frequentlyshortened, to geodesic, was preferred.[4]

This section treats the problem on an ellipsoid of revolu-tion (both oblate and prolate). The problem on a triaxialellipsoid is covered in the next section.When determining distances on the earth, various ap-proximate methods are frequently used; some of theseare described in the article on geographical distance.

1.1 Equations for a geodesic

Friedrich Bessel

4 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

d

-dR

d

a

b

R0

Fig. 2. Dierential element of a meridian ellipse.

+ d

ds dR d

N

d

Fig. 3. Dierential element of a geodesic on an ellipsoid.

Here the equations for a geodesic are developed; theseallow the geodesics of any length to be computed ac-curately. The following derivation closely follows thatof Bessel (1825). Bagratuni (1962, 15), Krakiwsky &Thomson (1974, 4), Rapp (1993, 1.2), and Borre &Strang (2012) also provide derivations of these equations.Consider an ellipsoid of revolution with equatorial radiusa and polar semi-axis b. Dene the attening f = (a b)/a, the eccentricity e2 = f(2 f), and the second ec-centricity e = e/(1 f). (In most applications in geodesy,the ellipsoid is taken to be oblate, a > b; however, the the-ory applies without change to prolate ellipsoids, a < b, inwhich case f, e2, and e2 are negative.)Let an elementary segment of a path on the ellipsoid havelength ds. From Figs. 2 and 3, we see that if its azimuthis , then ds is related to d and d by

: cosds = d = dR/ sin; sinds = Rd;where is the meridional radius of curvature, R = cosis the radius of the circle of latitude , and is the normalradius of curvature. The elementary segment is thereforegiven by

ds2 = 2 d2 +R2 d2

or

ds =p202 +R2 d

L(; 0) d;where = d/d and L depends on through () andR(). The length of an arbitrary path between (1, 1)and (2, 2) is given by

s12 =

Z 21

L(; 0) d;

where is a function of satisfying (1) = 1 and (2)= 2. The shortest path or geodesic entails nding thatfunction () which minimizes s12. This is an exercise inthe calculus of variations and the minimizing condition isgiven by the Beltrami identity,

L 0 @L@0

= const.

Substituting for L and using Eqs. (1) gives

0 aR = a cos

a

b

Z

a sin

P

P

Fig. 4. The geometric construction for parametric latitude, . Apoint P at latitude on the meridian (red) is mapped to a pointP on a sphere of radius a (shown as a blue circle) by keeping theradius R constant.

R sin = const.

Clairaut (1735) rst found this relation, using a geo-metrical construction; a similar derivation is presentedby Lyusternik (1964, 10).[5] Dierentiating this rela-tion and manipulating the result gives (Jekeli 2012, Eq.(2.95))

1.1 Equations for a geodesic 5

d = sind:

This, together with Eqs. (1), leads to a system of ordinarydierential equations for a geodesic (Borre & Strang2012, Eqs. (11.71) and (11.76))

:d

ds=

cos

;d

ds=

sin cos ;

d

ds=

tan sin

:

We can expressR in terms of the parametric latitude, ,[6]using

R = a cos

(see Fig. 4 for the geometrical construction), andClairauts relation then becomes

sin1 cos1 = sin2 cos2:

121

2

A

B

12

N12

0E

F H

Fig. 5. Geodesic problem mapped to the auxiliarysphere.

(s)

P

()

N( - 0)

0E G

Fig. 6. The elementary geodesic problem on the auxil-iary sphere.

This is the sine rule of spherical trigonometry relating twosides of the triangleNAB (see Fig. 5), NA = 1, andNB = 2 and their opposite angles B = 2 andA = 1.In order to nd the relation for the third side AB = 12,the spherical arc length, and included angle N = 12, thespherical longitude, it is useful to consider the triangleNEP representing a geodesic starting at the equator; see

Fig. 6. In this gure, the variables referred to the auxil-iary sphere are shown with the corresponding quantitiesfor the ellipsoid shown in parentheses. Quantities with-out subscripts refer to the arbitrary point P; E, the point atwhich the geodesic crosses the equator in the northwarddirection, is used as the origin for , s and .

+ d

ddcos d

N

d

Fig. 7. Dierential element of a geodesic on a sphere.

If the side EP is extended bymoving P innitesimally (seeFig. 7), we obtain

: cosd = d; sind = cos d!:

Combining Eqs. (1) and (3) gives dierential equationsfor s and

1

a

ds

d=

d

d!=

sinsin:

Up to this point, we have not made use of the specicequations for an ellipsoid, and indeed the derivation ap-plies to an arbitrary surface of revolution.[7] Bessel nowspecializes to an ellipsoid in which R and Z are related by

R2

a2+Z2

b2= 1;

where Z is the height above the equator (see Fig. 4). Dif-ferentiating this and setting dR/dZ = sin/cos gives

R sina2

Z cosb2

= 0;

eliminating Z from these equations, we obtain

R

a= cos = cosp

1 e2 sin2 :

This relation between and can be written as

6 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

tan =p1 e2 tan = (1 f) tan;

which is the normal denition of the parametric latitudeon an ellipsoid. Furthermore, we have

sinsin =

p1 e2 cos2 ;

so that the dierential equations for the geodesic become

1

a

ds

d=

d

d!=p1 e2 cos2 :

The last step is to use as the independent parameter[8] inboth of these dierential equations and thereby to expresss and as integrals. Applying the sine rule to the verticesE and G in the spherical triangle EGP in Fig. 6 gives

sin = sin(;0) = cos0 sin;

where 0 is the azimuth at E. Substituting this into theequation for ds/d and integrating the result gives

:

s

b=

Z 0

p1 e2 cos2 (0;0)

1 f d0

=

Z 0

p1 + k2 sin2 0 d0;

where

k = e0 cos0;

and the limits on the integral are chosen so that s( = 0) =0. Legendre (1811, p. 180) pointed out that the equationfor s is the same as the equation for the arc on an ellipsewith semi-axes b(1 + e2 cos20)1/2 and b. In order toexpress the equation for in terms of , we write

d! =sin0cos2 d;

which follows from Eq. (3) and Clairauts relation. Thisyields

:

0 = (1 f) sin0Z 0

p1 + k2 sin2 0

1 cos2 0 sin2 0d0

= ! sin0Z 0

e2

1 +p1 e2 cos2 (0;0)

d0

= ! f sin0Z 0

2 f1 + (1 f)

p1 + k2 sin2 0

d0;

and the limits on the integrals are chosen so that = 0 atthe equator crossing, = 0.In using these integral relations, we allow to increasecontinuously (not restricting it to a range [, ], for ex-ample) as the great circle, resp. geodesic, encircles theauxiliary sphere, resp. ellipsoid. The quantities , , ands are likewise allowed to increase without limit. Once theproblem is solved, can be reduced to the conventionalrange.This completes the solution of the path of a geodesic us-ing the auxiliary sphere. By this device a great circle canbe mapped exactly to a geodesic on an ellipsoid of revolu-tion. However, because the equations for s and in termsof the spherical quantities depend on 0, the mapping isnot a consistent mapping of the surface of the sphere tothe ellipsoid or vice versa; instead, it should be viewedmerely as a convenient tool for solving for a particulargeodesic.There are also several ways of approximating geodesicson an ellipsoid which usually apply for suciently shortlines (Rapp 1991, 6); however, these are typically com-parable in complexity to the method for the exact solutiongiven above (Jekeli 2012, 2.1.4).

1.2 Behavior of geodesics

N

E

Fig. 8. Meridians and the equator are the only closed geodesics.(For the very attened ellipsoids, there are other closed geodesics;see Figs. 13 and 14).

Geodesic on an oblate ellipsoid (f = 1/50) with 0 = 45.45o

0o

-45o

180o

360o

- 0

Fig. 9. Latitude as a function of longitude for a single cy-cle of the geodesic from one northward equatorial cross-ing to the next.

1.2 Behavior of geodesics 7

N

E

Fig. 10. Following the geodesic on the ellipsoid for about5 circuits.

Fig. 11. The same geodesic after about 70 circuits.Before solving for the geodesics, it is worth reviewing

N

E

Fig. 12. Geodesic on a prolate ellipsoid (f = 1/50) with 0 =45. Compare with Fig. 10.

their behavior. Fig. 8 shows the simple closed geodesicswhich consist of the meridians (green) and the equa-tor (red). (Here the qualication simple means thatthe geodesic closes on itself without an intervening self-

intersection.) This follows from the equations for thegeodesics given in the previous section.For meridians, we have 0 = 0 and Eq. (5) becomes = + 0, i.e., the longitude will vary the same way as for asphere, jumping by each time the geodesic crosses thepole. The distance, Eq. (4), reduces to the length of anarc of an ellipse with semi-axes a and b (as expected),expressed in terms of parametric latitude, .The equator ( = 0 on the auxiliary sphere, = 0 on theellipsoid) corresponds to 0 = . The distance reducesto the arc of a circle of radius b (and not a), s = b, whilethe longitude simplies to = (1 f) + 0. A geodesicthat is nearly equatorial will intersect the equator at inter-vals of b. As a consequence, the maximum length of aequatorial geodesic which is also a shortest path is b onan oblate ellipsoid (on a prolate ellipsoid, the maximumlength is a).All other geodesics are typied by Figs. 9 to 11. Figure9 shows latitude as a function of longitude for a geodesicstarting on the equator with 0 = 45. A full cycle ofthe geodesic, from one northward crossing of the equa-tor to the next, is shown. The equatorial crossings arecalled nodes and the points of maximum or minimumlatitude are called vertices; the vertex latitudes are givenby || = ( |0|). The latitude is an odd, resp.even, function of the longitude about the nodes, resp. ver-tices. The geodesic completes one full oscillation in lat-itude before the longitude has increased by 360. Thus,on each successive northward crossing of the equator (seeFig. 10), falls short of a full circuit of the equator by ap-proximately 2 f sin0 (for a prolate ellipsoid, this quan-tity is negative and completes more that a full circuit;see Fig. 12). For nearly all values of 0, the geodesicwill ll that portion of the ellipsoid between the two ver-tex latitudes (see Fig. 11).Two additional closed geodesics for the oblate ellipsoid,b/a = 2/7.

N

E

Fig. 13. Side view.

8 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

N

Fig. 14. Top view.

If the ellipsoid is suciently oblate, i.e., b/a < , anotherclass of simple closed geodesics is possible (Klingenberg1982, 3.5.19). Two such geodesics are illustrated inFigs. 13 and 14. Here b/a = 2/7 and the equatorial az-imuth, 0, for the green (resp. blue) geodesic is chosento be 53.175 (resp. 75.192), so that the geodesic com-pletes 2 (resp. 3) complete oscillations about the equatoron one circuit of the ellipsoid.

1.3 Evaluation of the integralsSolving the geodesic problems entails evaluating the in-tegrals for the distance, s, and the longitude, , Eqs. (4)and (5). In geodetic applications, where f is small, theintegrals are typically evaluated as a series; for this pur-pose, the second form of the longitude integral is pre-ferred (since it avoids the near singular behavior of therst form when geodesics pass close to a pole). In bothintegrals, the integrand is an even periodic function ofperiod . Furthermore, the term dependent on is mul-tiplied by a small quantity k2 = O(f). As a consequence,the integrals can both be written in the form

I = B0 +1Xj=1

Bj sin 2j

where B0 = 1 + O(f) and Bj = O(f j). Series expansionsfor Bj can readily be found and the result truncated sothat only terms which areO(f J) and larger are retained.[9](Because the longitude integral is multiplied by f, it is typ-ically only necessary to retain terms up to O(f J1) in thatintegral.) This prescription is followed by many authors(Legendre 1806) (Oriani 1806) (Bessel 1825) (Helmert1880) (Rainsford 1955) (Rapp 1993). Vincenty (1975a)uses J = 3 which provides an accuracy of about 0.1 mmfor theWGS84 ellipsoid. Karney (2013) gives expansionscarried out for J = 6 which suces to provide full doubleprecision accuracy for |f | 1/50. Trigonometric seriesof this type can be conveniently summed using Clenshawsummation.

In order to solve the direct geodesic problem, it is nec-essary to nd given s. Since the integrand in the dis-tance integral is positive, this problem has a unique root,which may be found using Newtons method, noting thatthe required derivative is just the integrand of the dis-tance integral. Oriani (1833) instead uses series reversionso that can be found without iteration; Helmert (1880)gives a similar series.[10] The reverted series convergessomewhat slower that the direct series and, if |f | > 1/100,Karney (2013, addenda) supplements the reverted serieswith one step of Newtons method to maintain accuracy.Vincenty (1975a) instead relies on a simpler (but slower)function iteration to solve for .It is also possible to evaluate the integrals (4) and (5) bynumerical quadrature (Saito 1970) (Saito 1979) (Sjberg& Shirazian 2012) or to apply numerical techniquesfor the solution of the ordinary dierential equations,Eqs. (2) (Kivioja 1971) (Thomas & Featherstone 2005)(Panou et al. 2013). Such techniques can be used forarbitrary attening f. However, if f is small, e.g., |f | 1/50, they do not oer the speed and accuracy of the se-ries expansions described above. Furthermore, for arbi-trary f, the evaluation of the integrals in terms of ellip-tic integrals (see below) also provides a fast and accuratesolution. On the other hand, Mathar (2007) has tackledthe more complex problem of geodesics on the surface ata constant altitude, h, above the ellipsoid by solving thecorresponding ordinary dierential equations, Eqs. (2)with [, ] replaced by [ + h, + h].

A. M. Legendre

Geodesics on an ellipsoid was an early application ofelliptic integrals. In particular, Legendre (1811) writesthe integrals, Eqs. (4) and (5), as

1.4 Solution of the direct problem 9

Arthur Cayley

:s

b= E(; ik);

:

= (1 f) sin0G(; cos2 0; ik)

= e02

p1 + e02

sin0H(;e02; ik);

where

tan =s

1 + e02

1 + k2 sin2 tan!;

and

G(; 2; k) =

Z 0

p1 k2 sin2

1 2 sin2 d

=k2

2F (; k) +

1 k

2

2

(; 2; k);

H(; 2; k) =

Z 0

cos2 (1 2 sin2 )

p1 k2 sin2

d

=1

2F (; k) +

1 1

2

(; 2; k);

and F(, k), E(, k), and (, 2, k), are incompleteelliptic integrals in the notation of DLMF (2010,

19.2(ii)).[11][12] The rst formula for the longitude inEq. (7) follows directly from the rst form of Eq. (5).The second formula in Eq. (7), due to Cayley (1870),is more convenient for calculation since the elliptic inte-gral appears in a small term. The equivalence of the twoforms follows from DLMF (2010, Eq. (19.7.8)). Fastalgorithms for computing elliptic integrals are given byCarlson (1995) in terms of symmetric elliptic integrals.Equation (6) is conveniently inverted using Newtonsmethod. The use of elliptic integrals provides a goodmethod of solving the geodesic problem for |f | > 1/50.[13]

1.4 Solution of the direct problem

The basic strategy for solving the geodesic problems onthe ellipsoid is to map the problem onto the auxiliarysphere by converting , , and s, to , and , to solvethe corresponding great-circle problem on the sphere, andto transfer the results back to the ellipsoid.In implementing this program, we will frequently need tosolve the elementary spherical triangle for NEP in Fig.6 with P replaced by either A (subscript 1) or B (subscript2). For this purpose, we can applyNapiers rules for quad-rantal triangles to the triangleNEP on the auxiliary spherewhich give

sin0 = sin cos = tan! cot;cos = cos cos! = tan0 cot;cos = cos! cos0 = cot tan;sin = cos0 sin = cot tan!;sin! = sin sin = tan tan0:

We can also stipulate that cos 0 and cos0 0.[14]Implementing this plan for the direct problem is straight-forward. We are given 1, 1, and s12. From 1 weobtain 1 (using the formula for the parametric latitude).We now solve the triangle problem with P = A and 1 and1 given to nd 0, 1, and 1.[15] Use the distance andlongitude equations, Eqs. (4) and (5), together with theknown value of 1, to nd s1 and 0. Determine s2 = s1 +s12 and invert the distance equation to nd 2. Solve thetriangle problem with P = B and 0 and 2 given to nd2, 2, and 2. Convert 2 to 2 and substitute 2 and2 into the longitude equation to give 2.[16]

The overall method follows the procedure for solving thedirect problem on a sphere. It is essentially the programlaid out by Bessel (1825),[17] Helmert (1880, 5.9), andmost subsequent authors.Intermediate points, way-points, on a geodesic can befound by holding 1 and 1 xed and solving the directproblem for several values of s12. Once the rst waypointis found, only the last portion of the solution (starting withthe determination of s2) needs to be repeated for eachnew value of s12.

10 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

1.5 Solution of the inverse problem

The ease with which the direct problem can be solved re-sults from the fact that given 1 and 1, we can immedi-ately nd 0, the parameter in the distance and longitudeintegrals, Eqs. (4) and (5). In the case of the inverseproblem, we are given 12, but we cannot easily relatethis to the equivalent spherical angle 12 because 0 isunknown. Thus, the solution of the problem requires that0 be found iteratively. Before tackling this, it is worthunderstanding better the behavior of geodesics, this time,keeping the starting point xed and varying the azimuth.Geodesics from a single point (f = 1/10, 1 = 30)

E

Fig. 15. Geodesics, geodesic circles, and the cut locus.0o 30o 60o 1 = 90o

120o

150o 180o

0o 45o 90o 135o 180o-60o

-30o

0o

30o

- 1

Fig. 16. The geodesics shown on a plate carre projec-tion.

0o0o

90o

90o

180o

180o

1

12

Fig. 17. 12 as a function of 1 for 1 = 30 and 2 =20.

Suppose point A in the inverse problem has 1 = 30and 1 = 0. Fig. 15 shows geodesics (in blue) emanat-ing A with 1 a multiple of 15 up to the point at which

they cease to be shortest paths. (The attening has beenincreased to 1/10 in order to accentuate the ellipsoidal ef-fects.) Also shown (in green) are curves of constant s12,which are the geodesic circles centered A. Gauss (1828)showed that, on any surface, geodesics and geodesic cir-cle intersect at right angles. The red line is the cut locus,the locus of points which have multiple (two in this case)shortest geodesics from A. On a sphere, the cut locus is apoint. On an oblate ellipsoid (shown here), it is a segmentof the circle of latitude centered on the point antipodal toA, = 1. The longitudinal extent of cut locus is ap-proximately 12 [ f cos1, + f cos1]. IfA lies on the equator, 1 = 0, this relation is exact andas a consequence the equator is only a shortest geodesicif |12| (1 f). For a prolateellipsoid, the cut locus is a segment of the anti-meridiancentered on the point antipodal to A, 12 = , and thismeans that meridional geodesics stop being shortest pathsbefore the antipodal point is reached.The solution of the inverse problem involves determin-ing, for a given point B with latitude 2 and longitude 2which blue and green curves it lies on; this determines 1and s12 respectively. In Fig. 16, the ellipsoid has beenrolled out onto a plate carre projection. Suppose 2 =20, the green line in the gure. Then as 1 is varied be-tween 0 and 180, the longitude at which the geodesic in-tersects = 2 varies between 0 and 180 (see Fig. 17).This behavior holds provided that |2| |1| (otherwise the geodesic does not reach 2 for somevalues of 1). Thus, the inverse problem may be solvedby determining the value 1 which results in the givenvalue of 12 when the geodesic intersects the circle =2.This suggests the following strategy for solving the inverseproblem (Karney 2013). Assume that the points A and Bsatisfy

: 1 0; j2j j1j ; 0 12 :(There is no loss of generality in this assumption, sincethe symmetries of the problem can be used to generateany conguration of points from such congurations.)

1. First treat the easy cases, geodesics which lie on ameridian or the equator. Otherwise...

2. Guess a value of 1.3. Solve the so-called hybrid geodesic problem, given

1, 2, and 1 nd 12, s12, and 2, correspondingto therst intersection of the geodesic with the circle = 2.

4. Compare the resulting 12 with the desired value andadjust 1 until the two values agree. This completesthe solution.

Each of these steps requires some discussion.

1.5 Solution of the inverse problem 11

1. For an oblate ellipsoid, the shortest geodesic lies on ameridian if either point lies on a pole or if 12 = 0 or .The shortest geodesic follows the equator if 1 = 2 = 0and |12| (1 f). For a prolateellipsoid, the meridian is no longer the shortest geodesicif 12 = and the points are close to antipodal (this willbe discussed in the next section). There is no longitudinalrestriction on equatorial geodesics.2. In most cases a suitable starting value of 1 is foundby solving the spherical inverse problem[14]

tan1 =cos2 sin!12

cos1 sin2 sin1 cos2 cos!12 ;

with 12 = 12. This may be a bad approximation if Aand B are nearly antipodal (both the numerator and de-nominator in the formula above become small); however,this may not matter (depending on how step 4 is handled).3. The solution of the hybrid geodesic problem is as fol-lows. It starts the same way as the solution of the directproblem, solving the triangle NEP with P = A to nd 0,1, 1, and 0.[18] Nownd 2 from sin2 = sin0/cos2,taking cos2 0 (corresponding to the rst, northward,crossing of the circle = 2). Next, 2 is given by tan2= tan2/cos2 and 2 by tan2 = tan2/sin0.[14] Fi-nally, use the distance and longitude equations, Eqs. (4)and (5), to nd s12 and 12.[19]

4. In order to discuss how 1 is updated, let us dene theroot-nding problem in more detail. The curve in Fig.17 shows 12(1; 1, 2) where we regard 1 and 2 asparameters and 1 as the independent variable. We seekthe value of 1 which is the root of

g(1) 12(1;1; 2) 12 = 0;

where g(0) 0 and g() 0. In fact, there is a uniqueroot in the interval 1 [0, ]. Any of a number of root-nding algorithms can be used to solve such an equation.Karney (2013) uses Newtons method, which requires agood starting guess; however it may be supplemented bya fail-safe method, such as the bisection method, to guar-antee convergence.An alternative method for solving the inverse problem isgiven by Helmert (1880, 5.13). Let us rewrite the Eq.(5) as

12 = !12 f sin0Z 21

2 f1 + (1 f)

p1 + k2 sin2 0

d0

= !12 f sin0I(1; 2;0):

Helmerts method entails assuming that 12 = 12, solv-ing the resulting problem on auxiliary sphere, and obtain-ing an updated estimate of 12 using

F. R. Helmert

!12 = 12 + f sin0I(1; 2;0):This xed point iteration is repeated until convergence.Rainsford (1955) advocates this method and Vincenty(1975a) adopted it in his solution of the inverse prob-lem. The drawbacks of this method are that conver-gence is slower than obtained using Newtons method (asdescribed above) and, more seriously, that the processfails to converge at all for nearly antipodal points. Ina subsequent report, Vincenty (1975b) attempts to curethis defect; but he is only partially successfulthe NGS(2012) implementation still includes Vincentys x stillfails to converge in some cases. Lee (2011) has com-pared 17 methods for solving the inverse problem againstthe method given by Karney (2013).The shortest distance returned by the solution of the in-verse problem is (obviously) uniquely dened. However,if B lies on the cut locus of A there are multiple azimuthswhich yield the same shortest distance. Here is a catalogof those cases:

1 = 2 (with neither point at a pole). If 1 =2, the geodesic is unique. Otherwise there are twogeodesics and the second one is obtained by inter-changing 1 and 2. (This occurs when 12 for oblate ellipsoids.)

12 = (with neither point at a pole). If 1 = 0 or, the geodesic is unique. Otherwise there are twogeodesics and the second one is obtained by negat-ing 1 and 2. (This occurs when 1 + 2 0 forprolate ellipsoids.)

A and B are at opposite poles. There are innitelymany geodesics which can be generated by varying

12 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

the azimuths so as to keep 1 + 2 constant. (Forspheres, this prescription applies when A and B areantipodal.)

1.6 Dierential behavior of geodesics

C. F. Gauss

E. B. Christoel

Various problems involving geodesics require knowingtheir behavior when they are perturbed. This is usefulin trigonometric adjustments (Ehlert 1993), determiningthe physical properties of signals which follow geodesics,etc. Consider a reference geodesic, parameterized by sthe length from the northward equator crossing, and asecond geodesic a small distance t(s) away from it. Gauss(1828) showed that t(s) obeys the Gauss-Jacobi equation

:d2t(s)

ds2= K(s)t(s);

where K(s) is the Gaussian curvature at s. The solution

d1 m12d1A B

dt1 M12dt1A B

Fig. 18. Denition of reduced length and geodesic scale.

may be expressed as the sum of two independent solutions

t(s2) = Cm(s1; s2) +DM(s1; s2)

where

m(s1; s1) = 0;dm(s1; s2)

ds2

s2=s1

= 1;

M(s1; s1) = 1;dM(s1; s2)

ds2

s2=s1

= 0:

We shall abbreviatem(s1, s2) =m12, the so-called reducedlength, and M(s1, s2) = M12, the geodesic scale.[20] Theirbasic denitions are illustrated in Fig. 18. Christoel(1869) made an extensive study of their properties. Thereduced length obeys a reciprocity relation,

m12 +m21 = 0:

Their derivatives are

dm12ds2

=M21;

dM12ds2

= 1M12M21m12

:

Assuming that points 1, 2, and 3, lie on the same geodesic,then the following addition rules apply (Karney 2013),

1.7 Geodesic map projections 13

m13 = m12M23 +m23M21;

M13 =M12M23 (1M12M21)m23m12

;

M31 =M32M21 (1M23M32)m12m23

:

The reduced length and the geodesic scale are compo-nents of the Jacobi eld.The Gaussian curvature for an ellipsoid of revolution is

K =1

=

(1 e2 sin2 )2b2

=b2

a4(1 e2 cos2 )2 :

Helmert (1880, Eq. (6.5.1.)) solved the Gauss-Jacobiequation for this case obtaining

m12/b =

q1 + k2 sin2 2 cos1 sin2

q1 + k2 sin2 1 sin1 cos2

cos1 cos2J(2) J(1)

;

M12 = cos1 cos2 +p1 + k2 sin2 2p1 + k2 sin2 1

sin1 sin2

sin1 cos2J(2) J(1)

p1 + k2 sin2 1

;

where

J() =

Z 0

k2 sin2 0p1 + k2 sin2 0

d0

= E(; ik) F (; ik):As we see from Fig. 18 (top sub-gure), the separationof two geodesics starting at the same point with azimuthsdiering by d1 is m12 d1. On a closed surface such asan ellipsoid, we expect m12 to oscillate about zero. In-deed, if the starting point of a geodesic is a pole, 1 =, then the reduced length is the radius of the circle oflatitude,m12 = a cos2 = a sin12. Similarly, for a merid-ional geodesic starting on the equator, 1 = 1 = 0, wehave M12 = cos12. In the typical case, these quantitiesoscillate with a period of about 2 in 12 and grow lin-early with distance at a rate proportional to f. In trigono-metric adjustments over small areas, it may be possibleto approximate K(s) in Eq. (9) by a constant K. In thislimit, the solutions for m12 and M12 are the same as fora sphere of radius 1/K, namely,

m12 = sin(pKs12)/

pK; M12 = cos(

pKs12):

To simplify the discussion of shortest paths in this para-graph we consider only geodesics with s12 > 0. The pointat which m12 becomes zero is the point conjugate to thestarting point. In order for a geodesic betweenA and B, of

length s12, to be a shortest path it must satisfy the Jacobicondition (Jacobi 1837) (Jacobi 1866, 6) (Forsyth 1927,2627) (Bliss 1916), that there is no point conjugate toA between A and B. If this condition is not satised, thenthere is a nearby path (not necessarily a geodesic) whichis shorter. Thus, the Jacobi condition is a local propertyof the geodesic and is only a necessary condition for thegeodesic being a global shortest path. Necessary and suf-cient conditions for a geodesic being the shortest pathare:

for an oblate ellipsoid, |12| ; for a prolate ellipsoid, |12| , if 0 0; if 0 =0, the supplemental condition m12 0 is required if|12| = .

The latter condition above can be used to determinewhether the shortest path is a meridian in the case ofa prolate ellipsoid with |12| = .The derivative required to solve the inverse method us-ing Newtons method, 12(1; 1, 2) / 1, is given interms of the reduced length (Karney 2013, Eq. (46)).

1.7 Geodesic map projectionsTwo map projections are dened in terms of geodesics.They are based on polar and rectangular geodesic coordi-nates on the surface (Gauss 1828). The polar coordinatesystem (r, ) is centered on some point A. The coordi-nates of another point B are given by r = s12 and = 1 and these coordinates are used to nd the projectedcoordinates on a plane map, x = r cos and y = r sin.The result is the familiar azimuthal equidistant projec-tion; in the eld of the dierential geometry of surfaces,it is called the exponential map. Due to the basic prop-erties of geodesics (Gauss 1828), lines of constant r andlines of constant intersect at right angles on the surface.The scale of the projection in the radial direction is unity,while the scale in the azimuthal direction is s12/m12.The rectangular coordinate system (x, y) uses a referencegeodesic dened by A and 1 as the x axis. The point (x,y) is found by traveling a distance s13 = x from A alongthe reference geodesic to an intermediate point C andthen turning counter-clockwise and traveling along ageodesic a distance s32 = y. IfA is on the equator and 1 =, this gives the equidistant cylindrical projection. If 1= 0, this gives the Cassini-Soldner projection. Cassinismap of France placed A at the Paris Observatory. Due tothe basic properties of geodesics (Gauss 1828), lines ofconstant x and lines of constant y intersect at right angleson the surface. The scale of the projection in the y direc-tion is unity, while the scale in the x direction is 1/M32.The gnomonic projection is a projection of the spherewhere all geodesics (i.e., great circles) map to straightlines (making it a convenient aid to navigation). Sucha projection is only possible for surfaces of constant

14 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION

Gaussian curvature (Beltrami 1865). Thus a projectionin which geodesics map to straight lines is not possiblefor an ellipsoid. However, it is possible to construct an el-lipsoidal gnomonic projection in which this property ap-proximately holds (Karney 2013, 8). On the sphere, thegnomonic projection is the limit of a doubly azimuthalprojection, a projection preserving the azimuths fromtwo points A and B, as B approaches A. Carrying outthis limit in the case of a general surface yields an az-imuthal projection in which the distance from the cen-ter of projection is given by = m12/M12. Even thoughgeodesics are only approximately straight in this projec-tion, all geodesics through the center of projection arestraight. The projection can then be used to give an it-erative but rapidly converging method of solving someproblems involving geodesics, in particular, nding theintersection of two geodesics and nding the shortest pathfrom a point to a geodesic.The Hammer retroazimuthal projection is a variation ofthe azimuthal equidistant projection (Hammer 1910). Ageodesic is constructed from a central point A to someother point B. The polar coordinates of the projection ofB are r = s12 and = 2 (which depends on theazimuth at B, instead of at A). This can be used to deter-mine the direction from an arbitrary point to some xedcenter. Hinks (1929) suggested another application: ifthe central point A is a beacon, such as the Rugby Clock,then at an unknown location B the range and the bearingto A can be measured and the projection can be used toestimate the location of B.

1.8 Envelope of geodesics

Geodesics from a single point (f = 1/10, 1 = 30)

Fig. 19. The envelope of geodesics from a point A at 1= 30.

A

B

1

2

3

4

N

Fig. 20. The four geodesics connecting A and a point B,2 = 26, 12 = 175.

The geodesics from a particular point A if continued pastthe cut locus form an envelope illustrated in Fig. 19. Herethe geodesics for which 1 is a multiple of 3 are shownin light blue. (The geodesics are only shown for their rstpassage close to the antipodal point, not for subsequentones.) Some geodesic circles are shown in green; theseform cusps on the envelope. The cut locus is shown in red.The envelope is the locus of points which are conjugateto A; points on the envelope may be computed by ndingthe point at which m12 = 0 on a geodesic (and Newtonsmethod can be used to nd this point). Jacobi (1891) callsthis star-like gure produced by the envelope an astroid.Outside the astroid two geodesics intersect at each point;thus there are two geodesics (with a length approximatelyhalf the circumference of the ellipsoid) between A andthese points. This corresponds to the situation on thesphere where there are short and long routes on agreat circle between two points. Inside the astroid fourgeodesics intersect at each point. Four such geodesics areshown in Fig. 20 where the geodesics are numbered in or-der of increasing length. (This gure uses the same posi-tion for A as Fig. 15 and is drawn in the same projection.)The two shorter geodesics are stable, i.e., m12 > 0, so thatthere is no nearby path connecting the two points which isshorter; the other two are unstable. Only the shortest line(the rst one) has 12 . All the geodesics are tangentto the envelope which is shown in green in the gure. Asimilar set of geodesics for the WGS84 ellipsoid is givenin this table (Karney 2011, Table 1):The approximate shape of the astroid is given by

x2/3 + y2/3 = 1

or, in parametric form,

x = cos3 ; y = sin3 :

The astroid is also the envelope of the family of lines

1.10 Software implementations 15

x

cos +y

sin = 1;

where is a parameter. (These are generated by the rodof the trammel of Archimedes.) This aids in nding agood starting guess for 1 for Newtons method for ininverse problem in the case of nearly antipodal points(Karney 2013, 5).The astroid is the (exterior) evolute of the geodesic circlescentered at A. Likewise, the geodesic circles are involutesof the astroid.

1.9 Area of a geodesic polygonA geodesic polygon is a polygonwhose sides are geodesics.The area of such a polygon may be found by rst comput-ing the area between a geodesic segment and the equa-tor, i.e., the area of the quadrilateral AFHB in Fig. 1(Danielsen 1989). Once this area is known, the area of apolygon may be computed by summing the contributionsfrom all the edges of the polygon.Here we develop the formula for the area S12 of AFHBfollowing Sjberg (2006). The area of any closed regionof the ellipsoid is

T =

ZdT =

Z1

Kcosd d;

where dT is an element of surface area and K is theGaussian curvature. Now the GaussBonnet theorem ap-plied to a geodesic polygon states

=

ZK dT =

Zcosd d;

where

= 2 Xj

j

is the geodesic excess and j is the exterior angle at vertexj. Multiplying the equation for by R22, where R2 is theauthalic radius, and subtracting this from the equation forT gives[21]

T = R22 +

Z 1

KR22

cosd d

= R22 +

Z b2

(1 e2 sin2 )2 R22

cosd d;

where the value ofK for an ellipsoid has been substituted.Applying this formula to the quadrilateral AFHB, notingthat = 2 1, and performing the integral over gives

S12 = R22(21)+b2

Z 21

1

2(1 e2 sin2 )+tanh1(e sin)

2e sin R22b2

sind;

where the integral is over the geodesic line (so that isimplicitly a function of ). Converting this into an integralover , we obtain

S12 = R22E12e2a2 cos0 sin0

Z 21

t(e02) t(k2 sin2 )e02 k2 sin2

sin2

d;

where

t(x) = 1 + x+p1 + x

sinh1pxpx

;

and the notation E12 = 2 1 is used for the geodesicexcess. The integral can be expressed as a series valid forsmall f (Danielsen 1989) (Karney 2013, 6 and adden-dum).The area of a geodesic polygon is given by summing S12over its edges. This result holds provided that the polygondoes not include a pole; if it does 2 R22 must be addedto the sum. If the edges are specied by their vertices,then a convenient expression for E12 is

tan E122

=sin 12 (2 + 1)cos 12 (2 1)

tan !122:

This result follows from one of Napiers analogies.

1.10 Software implementationsAn implementation of Vincentys algorithm in Fortranis provided by NGS (2012). Version 3.0 includes Vin-centys treatment of nearly antipodal points (Vincenty1975b). Vincentys original formulas are used in manygeographic information systems. Except for nearly an-tipodal points (where the inverse method fails to con-verge), this method is accurate to about 0.1 mm for theWGS84 ellipsoid (Karney 2011, 9).The algorithms given in Karney (2013) are included inGeographicLib (Karney 2015). These are accurate toabout 15 nanometers for the WGS84 ellipsoid. Imple-mentations in several languages (C++, C, Fortran, Java,JavaScript, Python, Matlab, and Maxima) are provided.In addition to solving the basic geodesic problem, this li-brary can return m12, M12, M21, and S12. The libraryincludes a command-line utility, GeodSolve, for geodesiccalculations. As of version 4.9.1, the PROJ.4 library forcartographic projections uses the C implementation forgeodesic calculations. This is exposed in the command-line utility, geod, and in the library itself.

16 2 GEODESICS ON A TRIAXIAL ELLIPSOID

The solution of the geodesic problems in terms of ellip-tic integrals is included in GeographicLib (in C++ only),e.g., via the -E option to GeodSolve. This method of so-lution is about 23 times slower than using series expan-sions; however it provides accurate solutions for ellipsoidsof revolution with b/a [0.01, 100] (Karney 2013, ad-denda).

2 Geodesics on a triaxial ellipsoid

Solving the geodesic problem for an ellipsoid of revolu-tion is, from the mathematical point of view, relativelysimple: because of symmetry, geodesics have a constantof the motion, given by Clairauts relation allowing theproblem to be reduced to quadrature. By the early 19thcentury (with the work of Legendre, Oriani, Bessel, etal.), there was a complete understanding of the proper-ties of geodesics on an ellipsoid of revolution.On the other hand, geodesics on a triaxial ellipsoid (with3 unequal axes) have no obvious constant of the motionand thus represented a challenging unsolved problemin the rst half of the 19th century. In a remarkable pa-per, Jacobi (1839) discovered a constant of the motionallowing this problem to be reduced to quadrature also(Klingenberg 1982, 3.5).[22][23]

2.1 Triaxial coordinate systems

Gaspard Monge

The key to the solution is expressing the problem in theright coordinate system. Consider the ellipsoid denedby

Charles Dupin

h =X2

a2+Y 2

b2+Z2

c2= 1;

where (X,Y,Z) are Cartesian coordinates centered on theellipsoid and, without loss of generality, a b c > 0.[24]A point on the surface is specied by a latitude and longi-tude. The geographical latitude and longitude (, ) aredened by

rhjrhj =

0@ cos coscos sinsin

1A :The parametric latitude and longitude (, ) are denedby

X = a cos0 cos0;Y = b cos0 sin0;Z = c sin0:Jacobi (1866, 2627) employed the ellipsoidal latitudeand longitude (, ) dened by

X = a cos!pa2 b2 sin2 c2 cos2 p

a2 c2 ;

Y = b cos sin!;

Z = c sinpa2 sin2 ! + b2 cos2 ! c2p

a2 c2 :

In the limit b a, becomes the parametric latitude foran oblate ellipsoid, so the use of the symbol is consistent

2.2 Jacobis solution 17

Fig. 21. Ellipsoidal coordinates.

with the previous sections. However, is dierent fromthe spherical longitude dened above.[25]

Grid lines of constant (in blue) and (in green) aregiven in Fig. 21. In contrast to (, ) and (, ), (, ) isan orthogonal coordinate system: the grid lines intersectat right angles. The principal sections of the ellipsoid,dened by X = 0 and Z = 0 are shown in red. The thirdprincipal section, Y = 0, is covered by the lines = 90and = 0 or 180. These lines meet at four umbilicalpoints (two of which are visible in this gure) where theprincipal radii of curvature are equal. Here and in theother gures in this section the parameters of the ellipsoidare a:b:c = 1.01:1:0.8, and it is viewed in an orthographicprojection from a point above = 40, = 30.The grid lines of the ellipsoidal coordinates may be inter-preted in three dierent ways

1. They are lines of curvature on the ellipsoid, i.e.,they are parallel to the directions of principal curva-ture (Monge 1796).

2. They are also intersections of the ellipsoid withconfocal systems of hyperboloids of one and twosheets (Dupin 1813, Part 5).

3. Finally they are geodesic ellipses and hyperbolas de-ned using two adjacent umbilical points (Hilbert &Cohn-Vossen 1952, p. 188). For example, the linesof constant in Fig. 21 can be generated with thefamiliar string construction for ellipses with the endsof the string pinned to the two umbilical points.

Conversions between these three types of latitudes andlongitudes and the Cartesian coordinates are simple alge-braic exercises.The element of length on the ellipsoid in ellipsoidal co-ordinates is given by

ds2

(a2 b2) sin2 ! + (b2 c2) cos2 =b2 sin2 + c2 cos2

a2 b2 sin2 c2 cos2 d2

+a2 sin2 ! + b2 cos2 !

a2 sin2 ! + b2 cos2 ! c2 d!2

and the dierential equations for a geodesic are

d

ds=

1q(a2 b2) sin2 ! + (b2 c2) cos2

pa2 b2 sin2 c2 cos2 pb2 sin2 + c2 cos2

cos;

d!

ds=

1q(a2 b2) sin2 ! + (b2 c2) cos2

pa2 sin2 ! + b2 cos2 ! c2pa2 sin2 ! + b2 cos2 !

sin;

d

ds=

1

((a2 b2) sin2 ! + (b2 c2) cos2 )3/2(a2 b2) cos! sin!

pa2 sin2 ! + b2 cos2 ! c2p

a2 sin2 ! + b2 cos2 !cos

+(b2 c2) cos sin

pa2 b2 sin2 c2 cos2 p

b2 sin2 + c2 cos2 sin

:

2.2 Jacobis solution

C. G. J. Jacobi

Jacobi showed that the geodesic equations, expressed inellipsoidal coordinates, are separable. Here is how he re-counted his discovery to his friend and neighbor Bessel(Jacobi 1839, Letter to Bessel),

The day before yesterday, I reduced toquadrature the problem of geodesic lines on

18 2 GEODESICS ON A TRIAXIAL ELLIPSOID

Joseph Liouville

J. G. Darboux

an ellipsoid with three unequal axes. They arethe simplest formulas in the world, Abelianintegrals, which become the well knownelliptic integrals if 2 axes are set equal.

Knigsberg, 28th Dec. '38.

The solution given by Jacobi (Jacobi 1839) (Jacobi 1866,

28) is

=

Z pb2 sin2 + c2 cos2 dp

a2 b2 sin2 c2 cos2 p(b2 c2) cos2 Z p

a2 sin2 ! + b2 cos2 ! d!pa2 sin2 ! + b2 cos2 ! c2

q(a2 b2) sin2 ! +

:

As Jacobi notes a function of the angle equals a func-tion of the angle . These two functions are just Abelianintegrals... Two constants and appear in the solution.Typically is zero if the lower limits of the integrals aretaken to be the starting point of the geodesic and the di-rection of the geodesics is determined by . However,for geodesics that start at an umbilical points, we have = 0 and determines the direction at the umbilical point.The constant may be expressed as

= (b2 c2) cos2 sin2 (a2 b2) sin2 ! cos2 ;

where is the angle the geodesic makes with lines of con-stant . In the limit b a, this reduces to sin cos =const., the familiar Clairaut relation. A nice derivation ofJacobis result is given by Darboux (1894, 583584)where he gives the solution found by Liouville (1846) forgeneral quadratic surfaces. In this formulation, the dis-tance along the geodesic, s, is found using

ds

(a2 b2) sin2 ! + (b2 c2) cos2 =pb2 sin2 + c2 cos2 dp

a2 b2 sin2 c2 cos2 p(b2 c2) cos2 =

pa2 sin2 ! + b2 cos2 ! d!p

a2 sin2 ! + b2 cos2 ! c2q(a2 b2) sin2 ! +

:

An alternative expression for the distance is

ds =

pb2 sin2 + c2 cos2

p(b2 c2) cos2 dp

a2 b2 sin2 c2 cos2

+

pa2 sin2 ! + b2 cos2 !

q(a2 b2) sin2 ! + d!p

a2 sin2 ! + b2 cos2 ! c2:

2.3 Survey of triaxial geodesics 19

2.3 Survey of triaxial geodesics

Circumpolar geodesics, 1 = 0, 1 = 90.

Fig. 22. 1 = 45.1.

Fig. 23. 1 = 87.48.

On a triaxial ellipsoid, there are only 3 simple closedgeodesics, the three principal sections of the ellipsoidgiven by X = 0, Y = 0, and Z = 0.[26] To survey the othergeodesics, it is convenient to consider geodesics which in-tersect the middle principal section, Y = 0, at right angles.Such geodesics are shown in Figs. 2226, which use thesame ellipsoid parameters and the same viewing direc-tion as Fig. 21. In addition, the three principal ellipsesare shown in red in each of these gures.If the starting point is 1 (90, 90), 1 = 0, and 1= 90, then > 0 and the geodesic encircles the ellip-soid in a circumpolar sense. The geodesic oscillatesnorth and south of the equator; on each oscillation it com-pletes slightly less that a full circuit around the ellipsoidresulting, in the typical case, in the geodesic lling thearea bounded by the two latitude lines = 1. Two ex-amples are given in Figs. 22 and 23. Figure 22 showspractically the same behavior as for an oblate ellipsoid ofrevolution (because a b); compare to Fig. 11. How-ever, if the starting point is at a higher latitude (Fig. 22)the distortions resulting from a b are evident. All tan-gents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at = 1 (Chasles 1846) (Hilbert & Cohn-Vossen 1952, pp.223224).

Transpolar geodesics, 1 = 90, 1 = 180.

Fig. 24. 1 = 39.9.

Fig. 25. 1 = 9.966.

If the starting point is 1 = 90, 1 (0, 180), and 1= 180, then < 0 and the geodesic encircles the ellipsoidin a transpolar sense. The geodesic oscillates east andwest of the ellipse X = 0; on each oscillation it completesslightly more that a full circuit around the ellipsoid re-sulting, in the typical case, in the geodesic lling the areabounded by the two longitude lines = 1 and = 180 1. If a = b, all meridians are geodesics; the eect of a b causes such geodesics to oscillate east and west. Twoexamples are given in Figs. 24 and 25. The constrictionof the geodesic near the pole disappears in the limit bc; in this case, the ellipsoid becomes a prolate ellipsoidand Fig. 24 would resemble Fig. 12 (rotated on its side).All tangents to a transpolar geodesic touch the confocaldouble-sheeted hyperboloid which intersects the ellipsoidat = 1.If the starting point is 1 = 90, 1 = 0 (an umbilicalpoint), and 1 = 135 (the geodesic leaves the ellipse Y= 0 at right angles), then = 0 and the geodesic repeat-edly intersects the opposite umbilical point and returnsto its starting point. However, on each circuit the angle atwhich it intersects Y = 0 becomes closer to 0 or 180 sothat asymptotically the geodesic lies on the ellipse Y = 0(Hart 1849) (Arnold 1989, p. 265). This is shown in Fig.26. Note that a single geodesic does not ll an area onthe ellipsoid. All tangents to umbilical geodesics touchthe confocal hyperbola which intersects the ellipsoid at

20 3 APPLICATIONS

Fig. 26. An umbilical geodesic, 1 = 90, 1 = 0, 1 = 135.

the umbilic points.Umbilical geodesic enjoy several interesting properties.

Through any point on the ellipsoid, there are twoumbilical geodesics.

The geodesic distance between opposite umbilicalpoints is the same regardless of the initial directionof the geodesic.

Whereas the closed geodesics on the ellipses X = 0and Z = 0 are stable (an geodesic initially close toand nearly parallel to the ellipse remains close to theellipse), the closed geodesic on the ellipse Y = 0,which goes through all 4 umbilical points, is expo-nentially unstable. If it is perturbed, it will swingout of the plane Y = 0 and ip around before return-ing to close to the plane. (This behavior may repeatdepending on the nature of the initial perturbation.)

If the starting point A of a geodesic is not an umbilicalpoint, then its envelope is an astroid with two cusps lyingon = 1 and the other two on = 1 + (Sinclair2003). The cut locus for A is the portion of the line =1 between the cusps (Itoh & Kiyohara 2004).(Panou 2013) gives a method for solving the inverse prob-lem for a triaxial ellipsoid by directly integrating the sys-tem of ordinary dierential equations for a geodesic.(Thus, it does not utilize Jacobis solution.)

3 ApplicationsThe direct and inverse geodesic problems no longer playthe central role in geodesy that they once did. Insteadof solving adjustment of geodetic networks as a two-dimensional problem in spheroidal trigonometry, theseproblem are now solved by three-dimensional methods

Karl Weierstrass

Henri Poincar

(Vincenty & Bowring 1978). Nevertheless, terrestrialgeodesics still play an important role in several areas:

21

for measuring distances and areas in geographic in-formation systems;

the denition of maritime boundaries (UNCLOS2006);

in the rules of the Federal Aviation Administrationfor area navigation (RNAV 2007);

the method of measuring distances in the FAI Sport-ing Code (FAI 2013).

By the principle of least action, many problems in physicscan be formulated as a variational problem similar to thatfor geodesics. Indeed, the geodesic problem is equiva-lent to the motion of a particle constrained to move onthe surface, but otherwise subject to no forces (Laplace1799a) (Hilbert & Cohn-Vossen 1952, p. 222). For thisreason, geodesics on simple surfaces such as ellipsoids ofrevolution or triaxial ellipsoids are frequently used as testcases for exploring new methods. Examples include:

the development of elliptic integrals (Legendre1811) and elliptic functions (Weierstrass 1861);

the development of dierential geometry (Gauss1828) (Christoel 1869);

methods for solving systems of dierential equationsby a change of independent variables (Jacobi 1839);

the study of caustics (Jacobi 1891);

investigations into the number and stability of peri-odic orbits (Poincar 1905);

in the limit c 0, geodesics on a triaxial ellipsoidreduce to a case of dynamical billiards;

extensions to an arbitrary number of dimensions(Knrrer 1980);

geodesic ow on a surface (Berger 2010, Chap. 12).

4 See also Geographical distance

Great-circle navigation

Geodesics

Geodesy

Meridian arc

Rhumb line

Vincentys formulae

5 Notes[1] Here 2 is the forward azimuth at B. Some authors calcu-

late the back azimuth instead; this is given by 2 .

[2] This prompted a courteous note by Oriani (1826) notinghis previous work, of which, presumably, Bessel was un-aware, and also a thinly veiled accusation of plagiarismfrom Ivory (1826) (his phrase was second-hand fromGermany), which resulted in an angry rebuttal by Bessel(1827).

[3] Clairaut (1735) uses the circumlocution perpendicularsto the meridian"; this refers to Cassinis proposed mapprojection for France (Cassini 1735) where one of the co-ordinates was the distance from the Paris meridian.

[4] Kummell (1883) attempted to introduce the wordbrachisthode for geodesic. This eort failed.

[5] Laplace (1799a) showed that a particle constrained tomove on a surface but otherwise subject to no forcesmoves along a geodesic for that surface. Thus, Clairautsrelation is just a consequence of conservation of angularmomentum for a particle on a surface of revolution. Asimilar proof is given by Bomford (1952, 8.06).

[6] In terms of , the element of distance on the ellipsoid isgiven by ds2 = (a2 sin22 + b2 cos2) d2 + a2 cos2 d2.

[7] It may be useful to impose the restriction that the surfacehave a positive curvature everywhere so that the latitudebe single valued function of Z.

[8] Other choices of independent parameter are possible. Inparticular many authors use the vertex of a geodesic (thepoint of maximum latitude) as the origin for .

[9] Nowadays, the necessary algebraic manipulations, ex-panding in a Taylor series, integration, and perform-ing trigonometric simplications, can be carrying usinga computer algebra system. Earlier, Levallois & Dupuy(1952) gave recurrence relations for the series in terms ofWallis integrals and Pittman (1986) describes a similarmethod.

[10] Legendre (1806, Art. 13) also gives a series for in termsof s; but this is not suitable for large distances.

[11] Despite the presence of i = 1, the elliptic integrals inEqs. (6) and (7) are real.

[12] Rollins (2010) obtains dierent, but equivalent, expres-sions in terms of elliptic integrals.

[13] It is also possible to express the integrals in terms of Jacobielliptic functions (Jacobi 1855) (Luther 1855) (Forsyth1896) (Thomas 1970, Appendix 1). Halphen (1888) givesthe solution for the complex quantities R exp(i) = X iY in terms of Weierstrass sigma and zeta functions. Thisform is of interest because the separate periods of lati-tude and longitude of the geodesic are captured in a singledoubly periodic function; see also Forsyth (1927, 75.)

[14] When solving for , , or using a formula for its tangent,the quadrant should be determined from the signs of thenumerator of the expression for the tangent, e.g., using theatan2 function.

22 6 REFERENCES

[15] If 1 = 0 and 1 = , the equation for 1 is indetermi-nate and 1 = 0 may be used.

[16] Because tan = sin0 tan, changes quadrants in stepwith . It is therefore straightforward to express 2 so that12 indicates how often and in what sense the geodesic hasencircled the ellipsoid.

[17] Bessel (1825) treated the longitude integral approximatelyin order to reduce the number of parameters in the equa-tion from two to one so that it could be tabulated conve-niently.

[18] If 1 = 2 = 0, take sin1 = sin1 = 0, consistent withthe relations (8); this gives 1 = 1 = .

[19] The ordering in relations (8) automatically results in 12> 0.

[20] Bagratuni (1962, 17) uses the term coecient of con-vergence of ordinates for the geodesic scale.

[21] Sjberg (2006) multiplies by b2 instead of R22. How-ever, this leads to a singular integrand (Karney 2011, 15).

[22] This section is adapted from the documentation for Ge-ographicLib (Karney 2015, Geodesics on a triaxial ellip-soid)

[23] Even though Jacobi and Weierstrass (1861) use terrestrialgeodesics as the motivation for their work, a triaxial ellip-soid approximates the Earth only slightly better than an el-lipsoid of revolution. A better approximation to the shapeof the Earth is given by the geoid. However, geodesics ona surface of the complexity of the geoid are partly chaotic(Waters 2011).

[24] This notation for the semi-axes is incompatible with thatused in the previous section on ellipsoids of revolution,where a and b stood for the equatorial radius and polarsemi-axis. Thus the corresponding inequalities are a = a b > 0 for an oblate ellipsoid and b a = a > 0 for aprolate ellipsoid.

[25] The limit b c gives a prolate ellipsoid with playingthe role of the parametric latitude.

[26] If c/a < , there are other simple closed geodesics similarto those shown in Figs. 13 and 14 (Klingenberg 1982,3.5.19).

6 References Arnold, V. I. (1989). Mathematical Methods ofClassical Mechanics. Translated by K. Vogtmann& A. Weinstein (2nd ed.). Springer-Verlag. ISBN978-0-387-96890-2. OCLC 4037141.

Bagratuni, G. V. (1967) [1962]. Course inSpheroidal Geodesy (PDF). OCLC 6150611. Trans-lation from Russian of (Moscow, 1962) by U.S. Air Force (FTD-MT-64-390)

Beltrami, E. (1865). Risoluzione del problema: Ri-portare i punti di una supercie sopra un piano inmodo che le linee geodetiche vengano rappresentateda linee rette [Mapping a surface to a plane so thatgeodesics are represented by straight lines]. Annalidi Matematica Pura ed Applicata, Series 1 (in Italian)7: 185204. doi:10.1007/BF03198517.

Berger, M. (2010). Geometry Revealed. Translatedby L. J. Senechal. Springer. doi:10.1007/978-3-540-70997-8. ISBN 978-3-540-70996-1.

Bessel, F. W. (2010) [1825]. . Translated by C. F.F. Karney & R. E. Deakin. The calculation of lon-gitude and latitude from geodesic measurements.Astronomische Nachrichten 331 (8): 852861.arXiv:0908.1824. doi:10.1002/asna.201011352.English translation of Astron. Nachr. 4, 241254(1825). Errata.

Bessel, F. W. (1827). "ber einen Aufsatzvon Ivory im Philosophical Magazine [Commentson a paper by Ivory in the Philosophical Mag-azine]. Astronomische Nachrichten (in German)5 (108): 177180. Bibcode:1826AN......5..177B.doi:10.1002/asna.18270051202.

Bliss, G. A. (1916). Jacobis condition for prob-lems of the calculus of variations in parametricform. Transactions of the American MathematicalSociety 17 (2): 195206. doi:10.1090/S0002-9947-1916-1501037-4 (free access).

Bomford, G. (1952). Geodesy. Oxford: Clarendon.OCLC 1396190.

Borre, K.; Strang, W. G. (2012). Algorithms forGlobal Positioning. Wellesley-Cambridge Press.ISBN 978-0-9802327-3-8. OCLC 795014501.Chapter 11, Geometry of the Ellipsoid.

Carlson, B. C. (1995). Numerical computation ofreal or complex elliptic integrals. Numerical Al-gorithms 10 (1): 1398. arXiv:math/9409227.Bibcode:1995NuAlg..10...13C.doi:10.1007/BF02198293.

Casper, P. W.; Bent, R. B. (1991). The eect ofthe Earths oblate spheroid shape on the accuracyof a time-of-arrival lightning ground strike locat-ing system (PDF). In Proceedings 1991 Interna-tional Aerospace and Ground Conference on Light-ning and Static Electricity, (Vol. 2).

Cassini, J. (1735). De la carte de la France et de laperpendiculaire a la mridienne de Paris [The mapof France and the perpendicular to the meridian ofParis]. Mmoires de l'Acadmie Royale des Sciencesde Paris 1733 (in French): 389405.

Cayley, A. (1870). On the geodesic lines on anoblate spheroid. Philosophical Magazine (4th ser.)

23

40: 329340. doi:10.1080/14786447008640411(inactive 2015-05-11).

Chasles, M. (1846). Sur les lignes godsiques etles lignes de courbure des surfaces du second degr"[Geodesic lines and the lines of curvature of the sur-faces of the second degree]. Journal de Mathma-tiques Pures et Appliques (in French) 11: 520.

Christoel, E. B. (1869). Allgemeine Theorie dergeodtischen Dreiecke [General theory of geodesictriangles]. Abhandlungen Kniglichen Akademie derWissenschaft zu Berlin (in German): 119176.

Clairaut, A. C. (1735). Dterminationgometrique de la perpendiculaire la mri-dienne trace par M. Cassini [Geometricaldetermination of the perpendicular to the meridiandrawn by Jacques Cassini]. Mmoires de l'AcadmieRoyale des Sciences de Paris 1733 (in French):406416.

Danielsen, J. S. (1989). The Area under theGeodesic. Survey Review 30 (232): 6166.doi:10.1179/003962689791474267.

Darboux, J. G. (1894). Leons sur la thoriegnrale des surfaces [Lessons on the general theoryof surfaces] (in French) 3. Paris: Gauthier-Villars.OCLC 8566228. PDF.

DLMF (2010). Olver, F. W. J.; Lozier, D. W.;Boisvert, R. F.; Clark, C. W., eds. NIST Handbookof Mathematical Functions. Cambridge Univ. Press.ISBN 978-0-521-19225-5. MR 2723248.

Dupin, P. C. F. (1813). Dveloppements deGomtrie [Developments in geometry] (in French).Paris: Courcier. OCLC 560800801.

Ehlert, D. (1993). Methoden der ellipsoidischenDreiecksberechnung [Methods for ellipsoidal trian-gulatioin] (Technical report). Reihe B: AngewandteGeodsie, Heft Nr. 292 (in German). DeutscheGeodtische Kommission. OCLC 257615376.

Euler, L. (1755). "lmens de la trigonomtriesphrodique tirs de la mthode des plus grandset plus petits [Elements of spheroidal trigonom-etry taken from the method of maxima and min-ima]. Mmoires de l'Acadmie Royale des Sciencesde Berlin 1753 (in French) 9: 258293. Figures.

FAI (2013). FAI Sporting Code (Technical report).Lausanne, Switzerland: Fdration AronautiqueInternationale. Section 7.3.1.1.

Forsyth, A. R. (1896). Geodesics on an oblatespheroid. Messenger of Mathematics 25: 81124.PDF.

Forsyth, A. R. (1927). Calculus of Variations. Cam-bridge Univ. Press. ISBN 978-1-107-64083-2.OCLC 250050479.

Gauss, C. F. (1902) [1828]. General Investigationsof Curved Surfaces of 1827 and 1825. Translatedby J. C. Morehead and A. M. Hiltebeitel. PrincetonUniv. Lib. OCLC 7824448. PDF. English transla-tion of Disquisitiones generales circa supercies cur-vas (Dieterich, Gttingen, 1828).

Halphen, G. H. (1888). Trait des Fonctions El-liptiques et de leurs Applications [A Treatise on El-liptic Functions and their Applications] (in French)2. Gauthier-Villars. OCLC 25356730. Chapter 6.PDF.

Hammer, E. (1910). Gegenazimutale Projek-tion [Retroazimuthal projection]. Petermanns Ge-ographische Mitteilungen (in German) 56 (3): 153155 + plate.

Hansen, P. A. (1865). Geodtische Untersuchun-gen [Geodetic investigations] (in German). Leipzig:Hirzel. OCLC 7687476.

Hart, A. S. (1849). Geometrical demonstration ofsome properties of geodesic lines. Cambridge andDublin Mathematical Journal 4: 8084.

Helmert, F. R. (1964) [1880]. Mathematical andPhysical Theories of Higher Geodesy 1. St. Louis:Aeronautical Chart and Information Center. OCLC17273288. English translation of Die Mathema-tischen und Physikalischen Theorieen der HherenGeodsie, Vol. 1 (Teubner, Leipzig, 1880).

Hilbert, D.; Cohn-Vossen, S. (1952). Geometry andthe Imagination. Translated by P. Nemenyi. NewYork: Chelsea. OCLC 301610346.

Hinks, A. R. (1929). A retro-azimuthal equidistantprojection of the whole sphere. The GeographicalJournal 73 (3): 245247. doi:10.2307/1784715.JSTOR 1784715.

Hutton, C. (1811). A Course of Mathematics inThree Volumes Composed for the Use of the RoyalMilitary Academy. London. p. 115. OCLC18031510.

Itoh, J.-I.; Kiyohara, K. (2004). The cut loci andthe conjugate loci on ellipsoids. Manuscripta Math-ematica 114 (2): 247264. doi:10.1007/s00229-004-0455-z.

Ivory, J. (1826). On the properties of aline of shortest distance traced on the sur-face of an oblate spheroid, Part 2. Philo-sophical Magazine Series 1 67 (337): 340352.doi:10.1080/14786442608674069.

24 6 REFERENCES

Jacobi, C. G. J. (1837). Zur Theorie derVariations-Rechnung und der Dierential-Gleichungen [The theory of the calculus ofvariations and of dierential equations]. Jour-nal fr die reine und angewandte Mathematik(Crelles Journal) (in German) 1837 (17): 6882.doi:10.1515/crll.1837.17.68.

Jacobi, C. G. J. (1839). Note von der geodtis-chen Linie auf einem Ellipsoid und den verschiede-nen Anwendungen einer merkwrdigen analytis-chen Substitution [The geodesic on an ellipsoid andvarious applications of a remarkable analytical sub-stitution]. Journal fr die reine und angewandteMathematik (Crelles Journal) (in German) 19 (19):309313. doi:10.1515/crll.1839.19.309. Letter toBessel, Dec. 28, 1838. French translation (1841).

Jacobi, C. G. J. (1855). Communicatedby E. Luther. Nouvelles formules deGodsie [New formulas for geodesy]. As-tronomische Nachrichten (in French) 41 (14):209217. Bibcode:1855AN.....41..209J.doi:10.1002/asna.18550411401.

Jacobi, C. G. J. (2009) [1866]. A. Clebsch, ed.Lectures on Dynamics. Translated by K. Balagan-gadharan. New Delhi: Hindustan Book Agency.ISBN 978-81-85931-91-3. MR 2569315. OCLC440645889. English translation ofVorlesungen berDynamik (Reimer, Berlin, 1866). Errata.

Jacobi, C. G. J. (1891). "ber die Curve, welchealle von einem Punkte ausgehenden geodtischenLinien eines Rotationsellipsoides berhrt [The en-velope of geodesic lines emanating from a singlepoint on an ellipsoid]. In K. T. W. Weierstrass (ed-itor). Jacobis Gesammelte Werke (in German) 7.Berlin: Reimer. pp. 7287. OCLC 630416023.Op. post., completed by F. H. A. Wangerin. PDF.

Jekeli, C. (2012), Geometric Reference Systems inGeodesy, Ohio State Univ.

Karney, C. F. F. (2009). GeographicLib. Version2009-03. The rst implementation of geodesic rou-tines in Geographiclib.

Karney, C. F. F. (2011). Geodesics on an ellipsoidof revolution. arXiv:1102.1215v1 [physics.geo-ph]. Errata.

Karney, C. F. F. (2013). Algorithms forgeodesics. Journal of Geodesy 87 (1): 4342.arXiv:1109.4448. Bibcode:2013JGeod..87...43K.doi:10.1007/s00190-012-0578-z (open access).Addenda.

Karney, C. F. F. (2015). GeographicLib. Version1.44.

Kim, J. J.; Burnside, W. D. (1986). Sim-ulation and analysis of antennas radiatingin a complex environment. IEEE Transac-tions on Antennas and Propagation 34 (4):554562. Bibcode:1986ITAP...34..554K.doi:10.1109/TAP.1986.1143838.

Kivioja, L. A. (1971). Computation ofgeodetic direct and indirect problems bycomputers accumulating increments fromgeodetic line elements. Bulletin Godsique99: 5563. Bibcode:1971BGeod..45...55K.doi:10.1007/BF02521679.

Klingenberg, W. P. A. (1982). Riemannian Geom-etry. de Gruyer. ISBN 978-3-11-008673-7. MR666697. OCLC 8476832.

Knrrer, H. (1980). Geodesics on the el-lipsoid. Inventiones Mathematicae 59 (2):119143. Bibcode:1980InMat..59..119K.doi:10.1007/BF01390041.

Krakiwsky, E. J.; Thomson, D. B. (1974), Geodeticposition computations (PDF), Dept. of Geodesyand Geomatics Engineering, Lecture Notes (39),Fredericton, N.B.: Univ. of New Brunswick,Bibcode:1974gpc..book.....K

Kummell, C. H. (1883). Alignment of curves onany surface, with special application to the ellip-soid. Bulletin of the Philosophical Society of Wash-ington 6: 123132. PDF.

Laplace, P. S. (1829) [1799a]. Treatise on CelestialMechanics 1. Translated by N. Bowditch. Boston:Hillard, Gray, Little, & Wilkins. OCLC 1294937.Book 1, 8.

Laplace, P. S. (1799b). Trait de Mcanique Cleste[Treatise on Celestial Mechanics] (in French) 2.Paris: Crapelet. p. 112. OCLC 25448952.

Lee, Y. C. (2011). " "[The accuracy analysis of methods to solve thegeodetic inverse problem]. Journal of the Ko-rean Society of Surveying, Geodesy, Photogramme-try and Cartography (in Korean) 29 (4): 329341.doi:10.7848/ksgpc.2011.29.4.329.

Legendre, A. M. (1806). Analyse des trianglestraces sur la surface d'un sphrode [Analysis ofspheroidal triangles]. Mmoires de l'Institut Nationalde France (in French) (1st semester): 130161.

Legendre, A. M. (1811). Exercices de Calcul In-tgral sur Divers Ordres de Transcendantes et surles Quadratures [Exercises in Integral Calculus] (inFrench). Paris: Courcier. OCLC 312469983.

Levallois, J. J.; Dupuy, M. (1952). Note sur le Calculdes Grandes Godsiques (Technical report). Paris:Institut Gographique National. Chapter 2.

25

Liouville, J. (1846). Sur quelques cas particulierso les quations du mouvement d'un point matrielpeuvent sintgrer [Special cases where the equa-tions of motion are integrable]. Journal de Math-matiques Pures et Appliques (in French) 11: 345378.

Lyusternik, L. (1964). Shortest Paths: VariationalProblems. Popular Lectures in Mathematics 13.Translated by P. Collins and R. B. Brown. NewYork: Macmillan. MR 0178386. OCLC 1048605.Translation from Russian of : (Moscow, 1955).

Lyusternik, L.; Schnirelmann, L. (1929). Sur leproblme de trois godsiques fermes sur les sur-faces de genre 0 [The problem of three closedgeodesics on surfaces of genus 0]. Comptes Rendusde l'Acadmie des Sciences de Paris (in French) 189:269271.

Luther, E. (1855). Jacobis Ableitung der inseinem Aufsatze: 'Solution nouvelle d'un problmede Godsie fondamental' enthaltenen Formeln[Jacobis derivation of the formulas in 'New so-lution of a fundamental problem of geodesy'].Astronomische Nachrichten (in German) 42 (2223): 337358. Bibcode:1856AN.....42..337J.doi:10.1002/asna.18550422201.

Mathar, R. J. (2007). Geodetic line at con-stant altitude above the ellipsoid. arXiv:0711.0642[math.MG].

Monge, G. (1850) [1796]. Sur les lignes de cour-bure de la surface de l'ellipsode [On the lines ofcurvature on the surface of the ellipsoid]. In J. Liou-ville (editor). Application de l'Analyse la Gome-trie (in French) (5th ed.). Paris: Bachelier. pp. 139160. OCLC 2829112. Figures.

Munk, W. H.; Forbes, A. M. G. (1989). GlobalOcean Warming: An Acoustic Measure?". Jour-nal of Physical Oceanography 19 (11): 17651778.Bibcode:1989JPO....19.1765M. doi:10.1175/1520-0485(1989)0192.0.CO;2.

National Geodetic Survey (2012). Geodesic Utili-ties: Inverse and Forward. Version 3.0.

Newton, I. (1848) [1687]. TheMathematical Princi-ples of Natural Philosophy. Translated by A. Motte.New York: Adee. Book 3, Proposition 19, Problem3, pp. 405409.

Oriani, B. (1806). Elementi di trigonometria sfer-oidica, Pt. 1 [Elements of spheroidal trigonome-try]. Memorie dell'Istituto Nazionale Italiano (in Ital-ian) 1 (1): 118198.

Oriani, B. (1808). Elementi di trigonometria sfer-oidica, Pt. 2 [Elements of spheroidal trigonome-try]. Memorie dell'Istituto Nazionale Italiano (in Ital-ian) 2 (1): 158.

Oriani, B. (1810). Elementi di trigonometria sfer-oidica, Pt. 3 [Elements of spheroidal trigonome-try]. Memorie dell'Istituto Nazionale Italiano (in Ital-ian) 2 (2): 158.

Oriani, B. (1826). Auszug aus einemBriefe des Herrn Oriani an den Herausge-ber [Excerpt from a letter to the Editor].Astronomische Nachrichten (in French) 4(94): 461466. Bibcode:1826AN......4..461O.doi:10.1002/asna.18260043201.

Oriani, B. (1833). Nota aggiunta agli elementidella trigonometria sferoidica [Note added to theelements of spheroidal trigonometry]. Memoriedell'Imperiale Regio Istituto del Regno Lombardo-Veneto (in Italian) 4: 325331.

Panou, G.; Delikaraoglou, D.; Korakitis, R. (2013).Solving the geodesics on the ellipsoid as a bound-ary value problem. Journal of Geodetic Sci-ence 3 (1): 4047. Bibcode:2013JGeoS...3...40P.doi:10.2478/jogs-2013-0007 (open access).

Panou, G. (2013). The geodesic boundary valueproblem and its solution on a triaxial ellipsoid.Journal of Geodetic Science 3 (3): 240249.Bibcode:2013JGeoS...3..240P. doi:10.2478/jogs-2013-0028 (open access).

Pittman, M. E. (1986). Precision direct and inversesolutions of the geodesic. Surveying and Mapping46 (1): 4754.

Poincar, H. (1905). Sur les lignes godsiquesdes surfaces convexes [Geodesics lines on con-vex surfaces]. Transactions of the American Math-ematical Society (in French) 6 (3): 237274.doi:10.2307/1986219. JSTOR 1986219. PDF.

Rainsford, H. F. (1955). Long geodesicson the ellipsoid. Bulletin godsique 37:1222. Bibcode:1955BGeod..29...12R.doi:10.1007/BF02527187.

Rapp, R. H. (1991), Geometric geodesy, part I, OhioState Univ.

Rapp, R. H. (1993), Geometric geodesy, part II,Ohio State Univ.

RNAV (2007). Order 8260.54A, The United StatesStandard for Area Navigation (PDF) (Technical re-port). Washington, D.C.: U.S. Federal Aviation Ad-ministration. Appendix 2.

26 7 EXTERNAL LINKS

Rollins, C. M. (2010). An integral for geodesiclength (PDF). Survey Review 42 (315): 2026.doi:10.1179/003962609X451663.

Saito, T. (1970). The computation of longgeodesics on the ellipsoid by non-series ex-panding procedure. Bulletin Godsique 98:341373. Bibcode:1970BGeod..44..341S.doi:10.1007/BF02522166.

Saito, T. (1979). The computation of longgeodesics on the ellipsoid through Gaussianquadrature. Journal of Geodesy 53 (2):165177. Bibcode:1979BGeod..53..165S.doi:10.1007/BF02521087.

Sinclair, R. (2003). On the last geometric state-ment of Jacobi. Experimental Mathematics 12 (4):477485. doi:10.1080/10586458.2003.10504515.

Sjberg, L. E. (2006). Determinationof areas on the plane, sphere and ellip-soid. Survey Review 38 (301): 583593.doi:10.1179/003962606780732100.

Sjberg, L. E.; Shirazian, M. (2012). Solv-ing the direct and inverse geodetic problems onthe ellipsoid by numerical integration. Jour-nal of Surveying Engineering 138 (1): 916.doi:10.1061/(ASCE)SU.1943-5428.0000061.

Thomas, C. M.; Featherstone, W. E. (2005). Vali-dation of Vincentys formulas for the geodesic usinga new fourth-order extension of Kiviojas formula.Journal of Surveying Engineering 131 (1): 2026.doi:10.1061/(ASCE)0733-9453(2005)131:1(20).

Thomas, P. D. (1970). Spheroidal Geodesics, Refer-ence Systems, & Local Geometry (Technical report).U.S. Naval Oceanographic Oce. SP-138.

UNCLOS (2006). A Manual on Technical Aspectsof the United Nations Convention on the Law ofthe Sea, 1982 (PDF) (Technical report) (4th ed.).Monaco: International Hydrographic Bureau.

Vincenty, T. (1975a). Direct and inverse solutionsof geodesics on the ellipsoid with application ofnested equations (PDF). Survey Review 23 (176):8893. doi:10.1179/sre.1975.23.176.88. Adden-dum: Survey Review 23 (180): 294 (1976).

Vincenty, T. (1975b). Geodetic inverse solution be-tween antipodal points (PDF) (Technical report).DMAAC Geodetic Survey Squadron. Retrieved2011-07-28.

Vincenty, T.; Bowring, B. R. (1978). Applicationof three-dimensional geodesy to adjustments of hor-izontal networks (PDF) (Technical report). NOAA.NOS NGS-13.

Waters, T. J. (2012). Regular and ir-regular geodesics on spherical harmonicsurfaces. Physica D: Nonlinear Phenom-ena 241 (5): 543552. arXiv:1112.3231.Bibcode:2012PhyD..241..543W.doi:10.1016/j.physd.2011.11.010.

Weierstrass, K. T. W. (1861). "ber diegeodtischen Linien auf dem dreiaxigen Ellipsoid[Geodesic lines on a triaxial ellipsoid]. Monats-bericht Kniglichen Akademie der Wissenschaft zuBerlin (in German): 986997. PDF.

7 External links Online geodesic bibliography, approximately 180books and articles on geodesics on ellipsoids to-gether with links to online copies.

Implementations of Vincenty (1975a) for oblate el-lipsoids: NGS implementation, includes modicationsdescribed in Vincenty (1975b).

NGS online tools. Online calculator from Geoscience Australia. Javascript implementations of solutions todirect problem and inverse problem.

Implementation of Karney (2013) for ellipsoids ofrevolution in Geographiclib (Karney 2015): GeographicLib web site for downloading li-brary and documentation.

GeodSolve(1), man page for a utility forgeodesic calculations.

An online version of GeodSolve. Planimeter(1), man page for a utility for cal-culating the area of geodesic polygons.

An online version of Planimeter. geod(1), man page for the PROJ.4 utility forgeodesic calculations.

Javascript utility for direct and inverse prob-lems and area calculations.

Drawing geodesics on Google Maps. Matlab implementation of the geodesic rou-tines (used for the gures for geodesics on el-lipsoids of revolution in this article).

The rst description of the geodesic algo-rithms from (Karney 2009).

Geodesics on a triaxial ellipsoid: Additional notes about Jacobis solution. Caustics on an ellipsoid.

27

8 Text and image sources, contributors, and licenses8.1 Text

Geodesics on an ellipsoid Source: https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid?oldid=680091074 Contributors: TakuyaMu-rata, Michael Devore, Bgwhite, Arthur Rubin, NeilN, Colonies Chris, Huon, Lambiam, Euhedral, David Eppstein, Billinghurst, Ck,JuanFox, Jdaloner, DEMcAdams, Fgnievinski, Yobot, Citation bot, Trappist the monk, JanetteDoe, MelioraCogito, Groupuscule, Dexbot,FireySixtySeven, Crapscourge, Lovehumorsex and Anonymous: 5

8.2 Images File:Alexis_Clairault.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a0/Alexis_Clairault.jpg License: Public domain

Contributors: [1] Original artist: Louis-Jacques Cathelin File:Arthur_Cayley.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/Arthur_Cayley.jpg License: Public domain

Contributors: http://www-groups.dcs.st-and.ac.uk/~{}history/PictDisplay/Cayley.html Original artist: ? File:Azimutalprojektion-schief_kl-cropped.png Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/

Azimutalprojektion-schief_kl-cropped.png License: CC BY-SA 3.0 Contributors: https://commons.wikimedia.org/wiki/File:Azimutalprojektion-schief_kl.jpg Original artist: Stefan Khn (Fotograf)

File:Barnaba_Oriani.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/Barnaba_Oriani.jpg License: Public domainContributors: via here Original artist: ?

File:Carl_Friedrich_Gauss.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/9b/Carl_Friedrich_Gauss.jpg License:Public domain Contributors: Gau-Gesellschaft Gttingen e.V. (Foto: A. Wittmann). Original artist: Gottlieb BiermannA. Wittmann (photo)

File:Carl_Jacobi.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/90/Carl_Jacobi.jpg License: Public domain Contribu-tors: http://www.sil.si.edu/digitalcollections/hst/scientific-identity/explore.htm (reworked) Original artist: Unknown

File:Charles_Dupin.jpeg Source: https://upload.wikimedia.org/wikipedia/commons/d/d0/Charles_Dupin.jpeg License: Public domainContributors:Original artist: Julien-Leopold Boilly

File:Circumpolar_geodesic_on_a_triaxial_ellipsoid_case_A.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/Circumpolar_geodesic_on_a_triaxial_ellipsoid_case_A.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Circumpolar_geodesic_on_a_triaxial_ellipsoid_case_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f3/Circumpolar_geodesic_on_a_triaxial_ellipsoid_case_B.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Closed_geodesics_on_an_ellipsoid_of_revolution.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/cb/Closed_geodesics_on_an_ellipsoid_of_revolution.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Construction_for_parametric_latitude.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/61/Construction_for_parametric_latitude.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Darboux.jpg Source: https://upload.wikimedia.org/wikipedia/commons/7/70/Darboux.jpg License: Public domain Contributors:Ernest Lebons bookGaston Darboux, published byGauthier-Villars in 1910. Original artist: J. Colomb-Grard;[1] 25, Bld Bonne-Nouvelle,Paris 2me

File:Definition_of_reduced_length_and_geodesic_scale.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/78/Definition_of_reduced_length_and_geodesic_scale.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Differential_element_of_a_geodesic_on_a_sphere.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/30/Differential_element_of_a_geodesic_on_a_sphere.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Differential_element_of_a_geodesic_on_an_ellipsoid.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Differential_element_of_a_geodesic_on_an_ellipsoid.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Differential_element_of_a_meridian_ellipse.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/Differential_element_of_a_meridian_ellipse.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:Elwin_Bruno_Christoffel.JPG Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Elwin_Bruno_Christoffel.JPG Li-cense: Public domain Contributors: Forrs: http://www-gap.dcs.st-and.ac.uk/~{}history/BiogIndex.html (School of Mathematics andStatistics University of St Andrews, Scotland) Original artist: User MIstvan on hu.wikipedia

File:Envelope_of_geodesics_on_an_oblate_ellipsoid.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1e/Envelope_of_geodesics_on_an_oblate_ellipsoid.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Ck

File:F-R_Helmert_1.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/F-R_Helmert_1.jpg License: Public domainContributors: ? Original artist: ?

File:Four_geo